Production Functions or Functions of Production Q=f(a, b, c, …………n, T) Supervised By :Dr. Bina Pandey Presented By: Ganesh Kumar Gupta Date: 17 th Jan 2013

Production Functions or Functions of Production  Definition: A production function refers to the functional relationship, under the given technology, between physical rates of inputs & output of a firm, per unit time.  Mathematically, Q=f(a, b, c, …………n, T)……………………..(1) where, Q= the physical quantity of output per unit of time. a, b, c, …………n, represents the quantities of various inputs per unit of time. T= the prevailing assumed constant state of technology.  Estimation of Production Functions:  On operation Basis: a.The short Run Production function where Q=f(V, F)…………………………………………………………(1a) b.The Long Run Production function where Q=f(V)…………………………………………………………….(1b)

 Estimation of Production Functions contd…  For Empirical measurement : on the basis of the availability of data & the purpose of enquiry, the production functions can be: a. Linear Production function. b. Quadratic Production function. c. Cubic Production function. d. Power Production function. e. Cobb-Douglas Production function. f. CES Production function.

Cobb-Dauglas Production Function  Proposed by Knut Wickshell ( ) and tested against statistical evidence by Charles Cobb & Poul Douglas in  Considering a simplified view of American economy( ), Cobb & Douglas arrived to an empirical relationship of determining production output by the amount of labor involved and the amount of capital invested ;ceters paribus.  The Production function was of the form: Q(L,K)= K α L β ( 2) where, Q= Total production (the momentary value of all goods produced in a year. ) L=Labor unit(the total number of person-hours worked in a year) K=Capital input(the monetary worth of all machinery, equipment & buildings) A=The technology constant or factor of productivity. “α” and “β” are the output elasticity of labor and capital resp.; these values are constant and determined by the technology. A

Properties of Cobb-Douglas Production Function  So, Cobb-Douglas is a basically power function of two variables.  Homogenous of degree (α+β).  The returns of scale is immediately revealed by the sum of the two parameters α and β i.e.  Isoquants are negatively sloped and convex to the origin.  MRTS LK is a function of input ratio. if (α+β)=1 implies Constant Returns To Scale. if (α+β)>1 implies Increasing Returns To Scale. if (α+β)<1 implies Decreasing Returns To Scale.  Elasticity of substitution is equal to 1

An Example: Application Of Cobb-Douglas Production Function Year ……………….1920 L ………… K ………… Q ………… Table: Economic data of the American Economy during the period Declaration: the year 1899 taken as the base year value assigned 100. the other years values expressed as percentages of the 1899 figures. next, Cobbs & Douglas used the method of least squares to fit the data into the table to function. Considering the function from equation (1), Q(L,K) = AK α L β let A=1.01, α=0.75, β=0.25 If the values of the year 1904 and 1920 are taken, Q(121,138 )= 1.01( )( ) = ??……………………year 1904 Q(194,407 )= 1.01( )( ) = ??…………………….year 1920 Which are quite close to the actual values 122 and 231 respectively

TWO TYPES OF PROBLEMS TYPE-I : Equation Given: Q(L,K)= AK α L β ……………………………from (2) let A=1.3, α=0.7, β=0.3 HINT: Formulae: APL=Q/L, MPL= α (Q/L) APK=Q/K, MPK= β (Q/K) For Example: A researcher estimated Cobb-Douglas production function for soft drink industry by collecting information on number of workers and plant size for the 25 soft drink bottling plants in a country. Solution: Given A=1.36 α=0.64, β=0.33 Cobb-Douglas Equation Q(L,K)= AK α L β ……………………………from(2) Log form of equation Log Q=Log A + Log K + Log L………………………………….(3) Anti-log of log 1.36=18.21 TYPE-II :Formation of Equation Find APL, MPL, APK, MPK ? For example say L=120 and K=60 So the equation is Q(L,K)= 18.21(L 0.64 )(K 0.33 )

CES Production Function  Elasticity Of Substitution:  Elasticity of substitution is the elasticity of the ratio of two inputs to a production function with respect to the ratio of their marginal products/outputs.  Constant Elasticity Of Substitution:  its refers to a particular type of aggregator function which combines two or more types of productive inputs into an aggregative quantity. This function exhibits constant elasticity of substitution.  CES Production Function:  the CES production function is a type of production function that displays constant elasticity of substitution.  the production technology has a constant percentage change in factor (e. g. labor & capital ) proportion due to a percentage change in marginal rate of technical substitution.

CES Production Function contd…  the two factor(capital & labor) CES production function was introduced by Solow and later made popular by Arrow, Chenery, Minhas and Solow.  The Equation is Q=A[α K -ρ +(1- α ) L -ρ ] -r/ρ ………………………..(4) where,  A(>0) is the efficiency parameter which represents the size of the production function.  α (0< α <1)is the distribution parameter which will help us explain relative factor shares.  ρ is the substitution parameter which will help us derive the elasticity of substitution and  r is the scale parameter which determines the degree of homogeneity.  Homogeneous of degree ‘r’.  The value of ρ is less than or equal to 1 and can be –infinity. The two extreme cases are when ρ=-1 and ρ= infinity.

CES Production Function contd… CASE-I: The Perfect Substitution when ρ =-1 : the isoquants are straight lines for the production function. CASE-II: No Substitution when ρ = infinity. : the isoquants are at right angles. CASE-III: Unit Elasticity of Substitution when ρ =0.(Cobb-Douglas PF) : the isoquants are negatively sloped convex to origin. Fig: Graph showing the curvesfor the different value of ρ

THAT’S ALL THANK YOU FOR ATTENTION Any Queries or Comments……?

 Managerial Economics By D. M. Mithani.  Google.com  Wikipedia, Cobb-Douglas. douglas References: